Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations.(English)Zbl 1046.34037

The authors consider the boundary value problem (*) $$x^{(2m)}(t)+(-1)^{m+1}f(x(t))=0$$, for $$0<t<1,$$ $$x^{(2i)}(0)=x^{(2i)}(1),$$ $$i=0,1,2,\dots ,m-1,$$ where $$f:\mathbb{R}\rightarrow [ 0,\infty)$$ is continuous. Let $$G_{m}(t,s)$$ be Green’s function for problem (*). Using Krasnosel’skij fixed-point theorem, they show that if there are constants $$0<a_{1}<a_{2}<\cdots <a_{2k}$$ and $$r_{1},r_{3},\dots ,r_{2k-1}\in (0,1/2)$$ such that (1) $$f(z)<a_{i}/\int_{0}^{1}G_{m}(1/2,s)\,ds$$ for $$z\in [ 0,a_{i}] ,$$ $$i=2,4,\dots ,2k,$$ and (2) $$f(z)>a_{i}/\int_{r_{i}}^{1-r_{i}}G_{m}(1/2,s)\,ds$$ for $$z\in [ a_{i}A_{m}(r_{i}),a_{i}]$$, $$i=1,3,\dots ,2k-1$$, where $$A_{m}(r)$$ are some functions, then the boundary value problem (*) has at least $$2k-1$$ symmetric positive solutions. Another similar result provides an even number of symmetric positive solutions. As a conclusion, the relationships between the results in this article and some recent related work by J. Henderson and H. B. Thompson [Proc. Am. Math. Soc. 128, No. 8, 2373–2379 (2000; Zbl 0949.34016)] are given.

MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Zbl 0949.34016
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References:

 [1] Nguyen Thanh Long, E. L. Ortiz, and A. Pham Ngoc Dinh, On the existence of a solution of a boundary value problem for a non-linear Bessel equation on an unbounded interval, Proc. Roy. Irish Acad. Sect. A 95 (1995), no. 2, 237 – 247. · Zbl 0853.34022 [2] Ravi P. Agarwal, Donal O’Regan, and Patricia J. Y. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic Publishers, Dordrecht, 1999. · Zbl 1157.34301 [3] Ravi P. Agarwal and Fu-Hsiang Wong, Existence of positive solutions for higher order boundary value problems, Nonlinear Stud. 5 (1998), no. 1, 15 – 24. · Zbl 0928.34016 [4] Richard I. Avery, John M. Davis, and Johnny Henderson, Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem, Electron. J. Differential Equations (2000), No. 40, 15. · Zbl 0958.34020 [5] J. Baxley and L. J. Haywood, Nonlinear boundary value problems with multiple solutions, Nonlinear Anal. 47 (2001), 1187-1198. · Zbl 1042.34517 [6] J. Baxley and L. J. Haywood, Multiple positive solutions of nonlinear boundary value problems, Dynam. Contin. Discrete Impuls. Systems, to appear. [7] Paul W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, Electron. J. Qual. Theory Differ. Equ. (2000), No. 2, 8. · Zbl 0948.34013 [8] Paul W. Eloe and Johnny Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations (1995), No. 03, approx. 8 pp., issn=1072-6691, review=\MR{1316529},. · Zbl 0814.34017 [9] L. H. Erbe, Shou Chuan Hu, and Haiyan Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), no. 3, 640 – 648. · Zbl 0805.34021 · doi:10.1006/jmaa.1994.1227 [10] Johnny Henderson and H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2373 – 2379. · Zbl 0949.34016 [11] M. A. Krasnosel$$^{\prime}$$skiĭ, Positive solutions of operator equations, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. [12] Ma Ruyun, Zhang Jihui, and Fu Shengmao, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997), no. 2, 415 – 422. · Zbl 0892.34009 · doi:10.1006/jmaa.1997.5639 [13] Patricia J. Y. Wong and Ravi P. Agarwal, Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput. 104 (1999), no. 1, 15 – 31. · Zbl 0933.65089 · doi:10.1016/S0096-3003(98)10045-0
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